On the well-posedness of the wave map problem in high dimensions. (English) Zbl 1085.58022

Nahmod, Andrea; Stefanov, Atanas; Uhlenbeck, Karen

Commun. Anal. Geom. 11, No. 1, 49-83 (2003).

Summary: We construct a gauge theoretic change of variables for the wave map from \(\mathbb{R} \times\mathbb{R}^n\) into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation \(-n\geq 4\) – for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and \(n\geq 4\).

MSC:

58J45	Hyperbolic equations on manifolds
58D25	Equations in function spaces; evolution equations
58J32	Boundary value problems on manifolds
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B33	Critical exponents in context of PDEs
35L70	Second-order nonlinear hyperbolic equations